1. Field of the Invention
This invention relates broadly to the hydrocarbon industry. More particularly, this invention relates methods and apparatus for predicting pore and fracture pressures of a surface formation. The invention has particular application to the casing of wells undertaken during the drilling of the well, although the application is not limited thereto.
2. State of the Art
To safely drill a deep well for hydrocarbon exploration or production, it is necessary to prevent formation fluids from flowing into the well. This is typically done by adjusting the density of the drilling mud so that the wellbore pressure is at all depths above the pressure of formation fluids (the pore pressure). On the other hand, the mud density cannot be so great as to cause hydraulic fracturing of the formation (the fracture pressure). The pore pressure and the fracture pressure gradients thus provide minimum and maximum values for the mud weight that define a mud weight window (See, Burgoyne, A. T., Jr., Millheim, K. K., Chenevert, M. E., and Young, F. S., Jr., Applied Drilling Engineering, SPE Textbook Series, vol. 2, 1991).
In deep drilling, it is generally not possible to choose a mud weight that keeps the wellbore pressure within the bounds imposed by the pore and fracture pressures over the entire depth range of the well, and it is necessary to set a number of intermediate casing strings to hydraulically isolate the formation. As seen in FIG. 1, these casing strings are set at depths defined on the basis of the estimated pore and fracture pressure gradients. For safe and cost-effective drilling, it is then important to have a method to estimate pore and fracture pressures before drilling and to update these estimates as the well is being drilled and new information is acquired.
In current practice, estimates of the pore and fracture pressures can be obtained from information on the variation in compressional wave velocity with depth (in turn obtained from surface seismic data and borehole measurements). These estimates can then be calibrated with pressure data acquired during drilling. Although these pore and fracture pressure estimates are recognized to be inaccurate, a major limitation of current practice is that there is no quantification of their uncertainty.
An approach commonly used to estimate pore and fracture pressures is based on measurements of compressional wave velocities, formation resitivities, or drilling penetration rates. The fundamental assumption of this approach is that anomalies over a normal trend in depth of these measurements are related to corresponding anomalies in pore and fracture pressures. For example, if elevated pore pressures are due to undercompaction of shales, the sediment porosities will be anomalously high and velocities anomalously low. For purposes of simplicity, the description of the invention herein will focus on the use of compressional wave velocity to estimate pore and fracture pressures, because velocity estimates are typically available before drilling from the processing of surface seismic data (see, e.g., Sayers, C. M., Johnson, G. M., and Denyer, G., xe2x80x9cPore pressure prediction from seismic tomographyxe2x80x9d, paper OTC 11984 presented at the 2000 Offshore Technology Conference, Houston, May 14, 2000) and can then be refined with measurements acquired while drilling such as sonic logs, vertical seismic profiles (VSPs), or seismic MWD (see, e.g., Esmersoy, C., Underhill, W., and Hawthorn, A., xe2x80x9cSeismic measurement while drilling: conventional borehole seismics on LWDxe2x80x9d, paper RR presented at the 2001 Annual Symposium of the Society of Professional Well Log Analysts, Houston, Jun. 17-20, 2001).
Most methods for pore and fracture pressure prediction start from Terzaghi""s effective stress principle, which states that all effects of stress on measurable quantities (such as compressional wave velocities) are a function of the effective stress "sgr"v(z), defined as
"sgr"v(z)=pover(z)xe2x88x92ppore(z),xe2x80x83xe2x80x83(1)
where ppore(z) is the pore pressure at depth z, and pover(z) is the pressure due to the overburden. The pressure due to the overburden pover(z) is defined by
pover(z)=g∫xcfx81(zxe2x80x2)dzxe2x80x2,xe2x80x83xe2x80x83(2)
where g is the acceleration of gravity, xcfx81(z) is bulk density, and the integration is carried out from the surface to depth z. A commonly used formula to predict pore pressure is set forth in Eaton, B. A., xe2x80x9cThe equation for geopressure prediction from well logsxe2x80x9d, paper SPE 5544 presented at the 50th annual fall meeting of the Society of Petroleum Engineers, Dallas, Sept. 28-Oct. 1, 1975:
ppore(z)=pover(z)xe2x88x92[pover(z)xe2x88x92pnorm(z)][xcex1(z)/xcex1norm(z)]xe2x80x9d,xe2x80x83xe2x80x83(3)
where pnorm(Z)=pwgz is the normal (hydrostatic) pore pressure (pw, is the water density), xcex1(z) the compressional wave velocity, and xcex1norm(z) the normal trend expected when there are no overpressures. As set forth in Sayers, C. M., Johnson, G. M., and Denyer, G., xe2x80x9cPore pressure prediction from seismic tomographyxe2x80x9d, paper OTC 11984 presented at the 2000 Offshore Technology Conference, Houston, May 1-4, 2000, this normal trend can be taken to be a linear increase in velocity with depth:
xcex1norm(z)=xcex10+bxcex1[zxe2x88x92z0]xe2x80x83xe2x80x83(4)
where xcex10 is the velocity at the mudline, bxcex1the velocity gradient in the normal trend, and z0 the depth of the mudline.
Eaton""s equation (3) essentially predicts that where the compressional wave velocity follows the normal trend, the pore pressure should be close to its normal, hydrostatic value. If the velocity becomes smaller than the normal trend, the pore pressure predicted by equation (3) increases from the hydrostatic value. To apply this method to any particular location, one should determine the variation of velocity and density in depth, the value of the coefficients as and bxcex1in the normal velocity trend, and the value of the exponent n in equation (3). Compressional wave velocity can be estimated from surface seismic data. Density can be estimated from a local trend (e.g., established from logs in nearby wells) or from a relationship between velocity and density such as Gardner""s law (see, Gardner, G. H. F., Gardner, L. W., and Gregory, A. R., xe2x80x9cFormation velocity and density: The diagnostic basis for stratigraphic trapsxe2x80x9d, Geonhysics, 39, p. 770-780, 1974):
xcfx81(z)=A xcex1(z)B.xe2x80x83xe2x80x83(5)
The coefficients A and B in (5) can be obtained by fitting Gardner""s law in a cross-plot of logged values of compressional velocity and density; an example of which is shown in FIG. 2. Of course, the value of density cannot be exactly predicted by compressional velocity with Gardner""s law as the density predicted from velocity using Gardner""s law will have a residual uncertainty which is denoted by xcex94p and is typically a few percent of the density value.
The coefficients xcex10 and bxcex1in the normal velocity trend can be estimated from the velocity trend at depth intervals known or assumed to be in normal pressure conditions. Eaton originally suggested that the exponent n should be around 3. In Bowers, G. L., xe2x80x9cPore pressure estimation from velocity data: Accounting for overpressure mechanisms besides undercompactionxe2x80x9d, SPE Drilling and Completion, p. 89-95, June 1995 (1995), however, it was noted that if overpressures are due to mechanisms other than undercompaction the appropriate value of n should be higher (up to about 5). The coefficients xcex10 and bxcex1and the exponent n can be calibrated by measurements of pore pressure or mud weights in the well being drilled (see, e.g., Bowers, 1995, and see Sayers et al., 2000, both cited above).
Widely used methods for fracture pressure predictions also start from Terzaghi""s effective stress principle (equation 1 above). The most common stress state is one where the minimum effective stress "sgr"h(z) is horizontal, and can be written as a function of the vertical effective stress:
"sgr"h(z)=k(z)"sgr"v(z),xe2x80x83xe2x80x83(6)
where k(z)xe2x89xa61 is a dimensionless effective stress ratio. As originally proposed by Hubbert, M. K., and Willis, D. G., xe2x80x9cMechanics of hydraulic fracturingxe2x80x9d, Trans. AIME, 210, p. 153-166, (1957), hydraulic fracturing should occur when the wellbore pressure is greater than the minimum horizontal effective stress, so the fracturing pressure pfrac(Z) can be written as
pfrac(z)=ppore(z)+k(z)[pover(z)xe2x88x92ppore(z)].xe2x80x83xe2x80x83(7)
Hubbert and Willis originally proposed a constant value of ⅓for the effective stress ratio, but subsequently others recognized that the effective stress ratio increases with depth (Matthews, W. R., and Kelly, J., xe2x80x9cHow to predict formation pressure and fracture gradient from electric and sonic logsxe2x80x9d, The Oil and Gas Journal, Feb. 20, 1967, p. 92-106; Pennebaker, E. S., xe2x80x9cAn engineering interpretation of seismic dataxe2x80x9d, paper SPE 2165 presented at the 43rd annual fall meeting of the Society of Petroleum Engineers, Houston, Sep. 29-Oct. 2, 1968; Eaton, B. A., xe2x80x9cFracture gradient prediction and its application in oilfield operationsxe2x80x9d, Journal of Petroleum Technology, October 1969, p. 1353-1360). Published plots of the variation of the effective stress ratio k(z) with depth are well represented by an exponential function (e.g., Zamora, M., xe2x80x9cNew method predicts gradient fracturexe2x80x9d, Petroleum Engineer International, Sept. 1989, p. 3847, 1989); a relationship for k(z) is best written (following Terzaghi""s effective stress principle) as a function of the vertical effective stress:
k(z)=k∞xe2x88x92[k∞xe2x88x92k0]exp{xe2x88x92[pover(z)xe2x88x92ppore(z)]/bk}xe2x80x83xe2x80x83(8)
where k0 is the value of the effective stress ratio at the mudline and bk a decay constant controlling how quickly k(z) increases with increasing vertical effective stress, approaching an asymptotic value of k∞as depth goes to infinity.
To apply this method to any particular location, one needs to determine the variation of overburden and pore pressure with depth and the coefficients needed to determine the variation of the effective stress ratio with depth. Values of these coefficients in different regions can be found in the published literature (see, e.g., Zamora, M., cited above). These coefficients can be calibrated by measurements of fracture pressures in the well being drilled, typically done in leak-off tests taken after setting a casing string (Burgoyne, A. T., Jr., Millheim, K. K., Chenevert, M. E., and Young, F. S., Jr., xe2x80x9cApplied Drilling Engineeringxe2x80x9d, SPE Textbook Series, vol. 2, 1991).
The pore and fracture pressure relationships (equations 3 and 7) can be used to make a deterministic prediction; i.e., a single prediction of both the pore pressure and the fracture pressure based on the most likely values of all inputs. The fundamental inputs to equations (3) and (7) are:
Compressional wave velocity in depth at the drilling location xcex1(z), which can be determined from measurements related to compressional wave velocities, such as surface seismics, vertical seismic profiles, seismic MWD (see, e.g., Esmersoy, C., Underhill, W., and Hawthorn, A., xe2x80x9cSeismic measurement while drilling: conventional borehole seismics on LWDxe2x80x9d, paper RR presented at the 2001 Annual Symposium of the Society of Professional Well Log Analysts, Houston, Jun. 17-20, 2001), and wireline or LWD sonic log data;
Bulk density in depth at the drilling location xcfx81(z), which can be determined from a local trend in the variation of density with depth, a relationship between the compressional wave velocity and density such as Gardner""s law, and wireline or LWD density log data;
Coefficients needed in the pore pressure relationship, which can be written as a vector cpore=(xcex10, bxcex1, n) and can be determined from measurements related to pore pressures, such as direct measurements of pore pressures and mud weights;
Coefficients needed in the fracture pressure relationship, which can be written as a vector cfrac=(k∞, k0, bk), and can be determined from measurements related to fracture pressures, such as direct measurements of fracture pressures carried out during leak-off tests.
Given the inputs above, a deterministic prediction of the pore and fracture pressures can be computed by: constructing a best estimate of compressional wave velocity in depth xcex1(z); constructing a best estimate of density in depth xcfx81(z); computing an overburden pressure profile pover(z) by integrating the density profile xcfx81(z); computing pore pressure ppore(z) from equation (3) using pover(z), the velocity profile xcex1(z), and coefficients in cpore; and computing fracture pressure pfrac(z) from equation (7) using ppore(z), pover(z), and coefficients in cfrac. By using most likely values for the velocity xcex1(z) and coefficients cpore and cfrac, the final result will be a best estimate for each of the profiles ppore(z) and pfrac(z).
This method of estimating pore and fracture pressures is primarily applicable if certain assumptions are met; for example, that anomalous pore pressures are due to undercompaction and that the volume being drilled is not under horizontal compression. Even when these assumptions are valid, however, a straightforward application is problematic, because the necessary inputs to equations (3) and (7) are not generally known accurately.
It is therefore an object of the invention to provide methods and apparatus for the prediction of formation parameters.
It is another object of the invention to provide methods and apparatus for the prediction of formation parameters which utilizes all information available for such prediction at the time of the prediction.
It is a further object of the invention to provide methods and apparatus for the prediction of formation parameters which takes into account the accuracy of all data available for use in such prediction at the time of the prediction.
It is an additional object of the invention to provide methods and apparatus for the prediction of formation parameters which also provide measures of uncertainty associated with the prediction.
Another object of the invention is to provide methods and apparatus for the prediction of pore and fracture pressures in an earth formation;
A further object of the invention is to provide methods and apparatus for the prediction of pore and fracture pressures in an earth formation which utilizes information obtained while drilling a borehole.
An additional object of the invention is to provide methods and apparatus for the prediction of pore and fracture pressures in an earth formation which utilizes all information available at the time of prediction, takes into account the accuracy of the information, and provides a measure of uncertainty associated with the prediction.
In accord with the objects of the invention which will be discussed in more detail below, a method of predicting values of formation parameters as a function of depth from a formation surface is provided and includes: generating an initial prediction of a profile of the formation parameters and uncertainties associated with the initial prediction using information available regarding the formation; and obtaining information related to the formation parameters during drilling of the formation and updating the uncertainties as a function of the first prediction and the information obtained in a recursive fashion. At any point in the drilling, the initial prediction and updated uncertainties may be used to generate numerous formation parameter profiles consistent with the data and uncertainties which are plotted to provide a probabilistic representation of the formation parameters.
In a preferred embodiment of the invention, known equations are used for finding initial values for formation parameters such as compressional velocity and density (which may be used to calculate pore pressure and fracture pressure). The uncertainties associated with the initial values are preferably quantified by using probability density functions (PDFs). To determine these PDFs, a Bayesian approach is utilized where xe2x80x9cprior PDFsxe2x80x9d describe uncertainty prior to obtaining additional information, and xe2x80x9cposterior PDFsxe2x80x9d account for the additional information acquired. As additional information is acquired, the posterior PDFs are redefined to include the new information. According to a preferred aspect of the invention, uncertainty in the formation parameters is quantified by sampling their posterior PDFs given all the data with a Markov Chain Monte Carlo algorithm which generates numerous formation parameter profiles consistent with the data and the computed Bayesian uncertainties. The numerous formation parameter profiles may then be plotted to provide a probabilistic representation (e.g., a histogram) of the formation parameters.
Additional objects and advantages of the invention will become apparent to those skilled in the art upon reference to the detailed description taken in conjunction with the provided figures.